The midpoint subdivision schemes form a class of subdivision schemes for arbitrary two-manifold meshes. It is observed that the midpoint subdivision surfaces are spline surfaces except for finitely many extraordinary points, which make the analysis of smoothness more difficult. D. Zorin
and P. Schröder
[Comput. Aided Geom. Des. 18, No. 5, 429–454 (2001; Zbl 0969.68155
smoothness of midpoint subdivision surfaces of degree 2 to 9. The authors develop here a geometric framework, which enables them to prove
continuity of midpoint subdivision surfaces of any degree greater than 1.